\documentclass{article}

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\begin{document}
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	\begin{align*}
		\Gamma(s) & = \int_{0}^{+\infty} e^{-x} x^{s-1} \d \ \ (s > 0)
		\\
		& = \int_{0}^{+\infty} -x^{s-1} \d[(e^{-x})]
		\\
		& = 0 - \int_{0}^{+\infty} e^{-x} \d[(-x^{s-1})]
		\\
		& = \int_{0}^{+\infty} e^{-x} \d[(x^{s-1})]
		\\
		& = \int_{0}^{+\infty} e^{-x} (s - 1) x^{s-2} \d
		\\
		& = (s - 1) \int_{0}^{+\infty} e^{-x} x^{s-2} \d
		\\
		& = (s - 1) \Gamma(s - 1)
		\\
		& = (s - 1) (s - 2) \Gamma(s - 2)
		\\
		& = (s - 1) (s - 2) \cdots \Gamma(1)
		\\
		& = (s - 1) (s - 2) \cdots 1
		\\
		& = (s - 1)!
	\end{align*}
\end{document}
